A deep learning algorithm to optimize the outcome
Deep learning models for regression tasks are quite difficult to train so I would suggest not to start with them. Instead I would start with one of the approaches below and maybe try to use deep learning afterwards.

A classical approach to the problem could be to analyze your optimization software and this would probably lead you to some deterministic algorithm.

Different approaches can be to treat your optimization software as a black box - give it a wide variety of inputs, note down the variables you are interested in (execution time, cutting results, etc.) and try to fit some sort or regressor on it. 

One option is to follow Kourosh's idea and formulate it as a classical optimization problem.

If you prefer to use machine learning tools than I'd suggest that you start with a simple model like linear regression just to verify that there is any signal in the data that you can use. Afterwards you can look at more powerful algorithms like xgboost, regression trees, etc.